Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

The reference solution comes from that given in sheet SSLV07 /89 of guide VPCS (considering in addition a transverse isotropic elastic matrix). The analytical expression for the solution is as follows:

Travel:

:math:`u=-\frac{{\nu }_{\text{NL}}\rho gxz}{{E}_{N}}`

:math:`v=-\frac{{\nu }_{\text{NL}}\rho gyz}{{E}_{N}}`

:math:`w=\frac{\rho g{z}^{2}}{2{E}_{N}}+\frac{\rho g{\nu }_{\text{NL}}}{2{E}_{N}}({x}^{2}+{y}^{2})-\frac{\rho g{L}^{2}}{2{E}_{N}}`

Constraints:

:math:`{\sigma }_{\mathrm{zz}}=\rho gz` :math:`{\sigma }_{\mathrm{zz}}={\sigma }_{\mathrm{yy}}={\sigma }_{\mathrm{xy}}={\sigma }_{\mathrm{yz}}={\sigma }_{\mathrm{zx}}=0`


.. image:: images/10001D9C0000124B000012660B4559D5A08A05E3.svg
    :width: 235
    :height: 237


.. _RefImage_10001D9C0000124B000012660B4559D5A08A05E3.svg:


Benchmark results
----------------------

Move points :math:`B`, :math:`C`, :math:`D`, :math:`E`, and :math:`X`.

Constraints :math:`{\sigma }_{\mathrm{zz}}` in :math:`A` and :math:`E`


Uncertainty about the solution
---------------------------

Exact analytical results.


Bibliographical references
---------------------------


1. TIMOSHENKO (S.P) Theory of elasticity - Paris - Librairie Polytechnique Ch. Béranger, p.279 to 282 (1961)


2. S.W. TSAI, H.T. HAHN - Introduction to composite materials. Technomic Publishing Company (1980).